The plurality system is the simplest means of determining the outcome of an election. To win, a candidate need only poll more votes than any other single opponent; he need not, as required by the majority formula, poll more votes than the combined opposition. The more candidates contesting a constituency seat, the greater the probability that the winning candidate will receive only a minority of the votes cast. Countries using the plurality formula for national legislative elections include Canada, Great Britain, India, and the United States. Countries with plurality systems usually have had two main parties.

Under the majority system, the party or candidate winning more than 50 percent of the vote in a constituency is awarded the contested seat. A difficulty in systems with the absolute-majority criterion is that it may not be satisfied in contests in which there are more than two candidates. Several variants of the majority formula have been developed to address this problem. In Australia the alternative, or preferential, vote is used in lower-house elections. Voters rank the candidates on an alternative-preference ballot. If a majority is not achieved by first-preference votes, the weakest candidate is eliminated, and that candidate’s votes are redistributed to the other candidates according to the second preference on the ballot. This redistributive process is repeated until one candidate has collected a majority of the votes. In France a double-ballot system is employed for National Assembly elections. If no candidate secures a majority in the first round of elections, another round is required. In the second round, only those candidates securing the votes of at least one-eighth of the registered electorate in the first round may compete, and the candidate securing a plurality of the popular vote in the second round is declared the winner. Some candidates eligible for the second round withdraw their candidacy and endorse one of the leading candidates. In contrast to the two-party norm of the plurality system, France has what some analysts have called a “two-bloc” system, in which the main parties of the left and the main parties of the right compete against each other in the first round of an election to be the representative of their respective ideological group and then ally with one another to maximize their bloc’s representation in the second round. An infrequently used variant is the supplementary-vote system, which was instituted for London mayoral elections. Under this system, voters rank their top two preferences; in the event that no candidate wins a majority of first-preference votes, all ballots not indicating the top two vote getters as either a first or a second choice are discarded, and the combination of first and second preferences is used to determine the winner. Majority formulas usually are applied only within single-member electoral constituencies.

The majority and the plurality formulas do not always distribute legislative seats in proportion to the share of the popular vote won by the competing parties. Both formulas tend to reward the strongest party disproportionately and to handicap weaker parties, though these parties may escape the inequities of the system if their support is regionally concentrated. For example, in national elections in Britain in 2001, the Labour Party captured more than three-fifths of the seats in the House of Commons, even though it won barely two-fifths of the popular vote; in contrast, the Conservative Party won one-fourth of the seats with nearly one-third of the vote. Third-party representation varied considerably; whereas the Liberal Democrats, whose support was spread throughout the country, captured 8 percent of the seats with more than 18 percent of the vote, the Plaid Cymru, whose support is concentrated wholly in Wales, won 0.7 percent of the vote and 0.7 percent of the seats. The plurality formula usually, though not always, distorts the distribution of seats more than the majority system.

Proportional representation

Proportional representation requires that the distribution of seats broadly be proportional to the distribution of the popular vote among competing political parties. It seeks to overcome the disproportionalities that result from majority and plurality formulas and to create a representative body that reflects the distribution of opinion within the electorate. Because of the use of multimember constituencies in proportional representation, parties with neither a majority nor a plurality of the popular vote can still win legislative representation. Consequentially, the number of political parties represented in the legislature often is large; for example, in Israel there are usually more than 10 parties in the Knesset.

Although approximated in many systems, proportionality can never be perfectly realized. Not surprisingly, the outcomes of proportional systems usually are more proportional than those of plurality or majority systems. Nevertheless, a number of factors can generate disproportional outcomes even under proportional representation. The single most important factor determining the actual proportionality of a proportional system is the “district magnitude”—that is, the number of candidates that an individual constituency elects. The larger the number of seats per electoral district, the more proportional the outcome. A second important factor is the specific formula used to translate votes into seats. There are two basic types of formula: single transferable vote and party-list proportional representation.

Single transferable vote

Developed in the 19th century in Denmark and in Britain, the single transferable vote formula—or Hare system, after one of its English developers, Thomas Hare—employs a ballot that allows the voter to rank candidates in order of preference. When the ballots are counted, any candidate receiving the necessary quota of first preference votes—calculated as one plus the number of votes divided by the number of seats plus one—is awarded a seat. In the electoral calculations, votes received by a winning candidate in excess of the quota are transferred to other candidates according to the second preference marked on the ballot. Any candidate who then achieves the necessary quota is also awarded a seat. This process is repeated, with subsequent surpluses also being transferred, until all the remaining seats have been awarded. Five-member constituencies are considered optimal for the operation of the single transferable vote system.

Because it involves the aggregation of ranked preferences, the single transferable vote formula necessitates complex electoral computations. This complexity, as well as the fact that it limits the influence of political parties, probably accounts for its infrequent use; it has been used in Northern Ireland, Ireland, and Malta and in the selection of the Australian and South African senates. The characteristic of the Hare formula that distinguishes it from other proportional representation formulas is its emphasis on candidates, not parties. The party affiliation of the candidates has no bearing on the computations. The success of minor parties varies considerably; small centrist parties usually benefit from the vote transfers, but small extremist parties usually are penalized.

Party-list proportional representation

The basic difference between the single transferable vote formula and list systems—which predominate in elections in western Europe and Latin America—is that, in the latter, voters generally choose among party-compiled lists of candidates rather than among individual candidates. Although voters may have some limited choice among individual candidates, electoral computations are made on the basis of party affiliation, and seats are awarded on the basis of party rather than candidate totals. The seats that a party wins are allocated to its candidates in the order in which they appear on the party list. Several types of electoral formulas are used, but there are two main types: largest-average and greatest-remainder formulas.

In the largest-average formula, the available seats are awarded one at a time to the party with the largest average number of votes as determined by dividing the number of votes won by the party by the number of seats the party has been awarded plus a certain integer, depending upon the method used. Each time a party wins a seat, the divisor for that party increases by the same integer, which thus reduces its chances of winning the next seat. Under all methods, the first seat is awarded to the party with the largest absolute number of votes, since, no seats having been allocated, the average vote total as determined by the formula will be largest for this party. Under the d’Hondt method, named after its Belgian inventor, Victor d’Hondt, the average is determined by dividing the number of votes by the number of seats plus one. Thus, after the first seat is awarded, the number of votes won by that party is divided by two (equal to the initial divisor plus one), and similarly for the party awarded the second seat, and so on. Under the so-called Sainte-Laguë method, developed by Andre Sainte-Laguë of France, only odd numbers are used. After a party has won its first seat, its vote total is divided by three; after it wins subsequent seats, the divisor is increased by two. The d’Hondt formula is used in Austria, Belgium, Finland, and the Netherlands, and the Sainte-Laguë method is used in Denmark, Norway, and Sweden.

The d’Hondt formula has a slight tendency to overreward large parties and to reduce the ability of small parties to gain legislative representation. In contrast, the Sainte-Laguë method reduces the reward to large parties, and it generally has benefited middle-size parties at the expense of both large and small parties. Proposals have been made to divide lists by fractions (e.g., 1.4, 2.5, etc.) rather than integers to provide the most proportional result possible.

The greatest-remainder method first establishes a quota that is necessary for a party to receive representation. Formulas vary, but they are generally some variation of dividing the total vote in the district by the number of seats. The total popular vote won by each party is divided by the quota, and a seat is awarded as many times as the party total contains the full quota. If all the seats are awarded in this manner, the election is complete. However, such an outcome is unlikely. Seats that are not won by full quotas subsequently are awarded to the parties with the largest remainder of votes after the quota has been subtracted from each party’s total vote for each seat it was awarded. Seats are distributed sequentially to the parties with the largest remainder until all the district’s allocated seats have been awarded.

Minor parties generally fare better under the greatest-remainder formula than under the largest-average formula. The greatest-remainder formula is used in Israel and Luxembourg and for some seats in the Danish Folketing. Prior to 1994 Italy used a special variant of the greatest-remainder formula, called the Imperiali formula, whereby the electoral quota was established by dividing the total popular vote by the number of seats plus two. This modification increased the legislative representation of small parties but led to a greater distortion of the proportional ideal.

The proportionality of outcomes also can be diluted by the imposition of an electoral threshold that requires a political party to exceed some minimum percentage of the vote to receive representation. Designed to limit the political success of small extremist parties, such thresholds can constitute significant obstacles to representation. The threshold varies by country, having been set at 4 percent in Sweden, 5 percent in Germany, and 10 percent in Turkey.

Hybrid systems

In some countries, the majoritarian and proportional systems are combined into what are called mixed-member proportional or additional-members systems. Although there are a number of variants, all mixed-member proportional systems elect some representatives by proportional representation and the remainder by a nonproportional formula. The classic example of the hybrid system is the German Bundestag, which combines the personal link between representatives and voters with proportionality. The German constitution provides for the election of half the country’s parliamentarians by proportional representation and half by simple plurality voting in single-member constituencies. Each voter casts two ballots. The first vote (Erstimme) is cast for an individual to represent a constituency (Wahlkreise); the candidate receiving the most votes wins the election. The second vote (Zweitstimme) is cast for a regional party list. The results of the second vote determine the overall political complexion of the Bundestag. All parties that receive at least 5 percent of the national vote—or win at least three constituencies—are allocated seats on the basis of the percentage of votes that they receive. The votes of parties not receiving representation are reapportioned to the larger parties on the basis of their share of the vote. During the 1990s, a number of countries adopted variants of the German system, including Italy, Japan, New Zealand, and several eastern European countries (e.g., Hungary, Russia, and Ukraine). A hybrid system also was adopted by the British government for devolved assemblies in Scotland and Wales. One of the chief differences between mixed-member systems is the percentage of seats allocated by proportional and majoritarian methods. For example, in Italy and Japan, respectively, roughly three-fourths and three-fifths of all seats are apportioned through constituency elections.

A country’s choice of electoral system, like its conception of representation, generally reflects its particular cultural, social, historical, and political circumstances. Majority or plural methods of voting are most likely to be acceptable in relatively stable political cultures. In such cultures, fluctuations in electoral support from one election to the next reduce polarization and encourage political centrism. Thus, the “winner take all” implications of the majority or plurality formulas are not experienced as unduly deprivational or restrictive. In contrast, proportional representation is more likely to be found in societies with traditional ethnic, linguistic, and religious cleavages or in societies that have experienced class and ideological conflicts.

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Alternatives are the essence of elections. The word election is derived from the Latin verb legere, meaning "to choose." Elections are the process through which citizens choose who will represent them in government or what will be done about a particular issue. It is important, however, to distinguish between the form and the substance of elections. In some countries elections are held but voters do not have a free and genuine choice between at least two alternatives. Though most countries hold elections in at least the formal sense, in many of them the elections are not competitive. For instance, all but one party may be forbidden to participate, or certain opposition groups may be prohibited from running candidates for office. (See also voting.)